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Interatomic Coulombic Decay Widths of Helium Trimer: a Diatomics-In Interatomic Coulombic decay widths of helium trimer: A diatomics-in-molecules approach Nicolas Sisourat, Sévan Kazandjian, Aurélie Randimbiarisolo, and Přemysl Kolorenč Citation: The Journal of Chemical Physics 144, 084111 (2016); doi: 10.1063/1.4942483 View online: http://dx.doi.org/10.1063/1.4942483 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/144/8?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Interatomic Coulombic decay widths of helium trimer: Ab initio calculations J. Chem. Phys. 143, 224310 (2015); 10.1063/1.4936897 Calculation of interatomic decay widths of vacancy states delocalized due to inversion symmetry J. Chem. Phys. 125, 094107 (2006); 10.1063/1.2244567 Interatomic Coulombic decay in a heteroatomic rare gas cluster J. Chem. Phys. 124, 154305 (2006); 10.1063/1.2185637 Observation of resonant Interatomic Coulombic Decay in Ne clusters J. Chem. 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Downloaded to IP: 195.113.23.13 On: Thu, 25 Feb 2016 16:39:07 THE JOURNAL OF CHEMICAL PHYSICS 144, 084111 (2016) Interatomic Coulombic decay widths of helium trimer: A diatomics-in-molecules approach Nicolas Sisourat,1 Sévan Kazandjian,1 Aurélie Randimbiarisolo,1,a) and Přemysl Kolorenč2 1Sorbonne Universités, UPMC University Paris 06, CNRS, Laboratoire de Chimie Physique Matière et Rayonnement, F-75005 Paris, France 2Charles University in Prague, Faculty of Mathematics and Physics, Institute of Theoretical Physics, V Holešoviˇckách 2, 180 00 Prague, Czech Republic (Received 17 December 2015; accepted 9 February 2016; published online 25 February 2016) We report a new method to compute the Interatomic Coulombic Decay (ICD) widths for large clusters which relies on the combination of the projection-operator formalism of scattering theory and the diatomics-in-molecules approach. The total and partial ICD widths of a cluster are computed from the energies and coupling matrix elements of the atomic and diatomic fragments of the system. The method is applied to the helium trimer and the results are compared to fully ab initio widths. A good agreement between the two sets of data is shown. Limitations of the present method are also discussed. C 2016 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4942483] I. INTRODUCTION phenomenon. Several computational approaches have been developed and used to calculate ICD widths.26–29 Among Resonance phenomena appear in many physical, chem- them, methods based on Complex Absorbing Potential (CAP) ical, and biological processes,1 for example, in atom- provide only total widths. Partial widths can be obtained molecule or electron-molecule collisions, photoionization, and using Wigner-Weisskopf method. However, this approach is autoionization events. Even though resonance states have been based on the lowest-order perturbation theory and thus gives investigated for decades,2 computing their properties is still a only estimates for the widths. To date, only the Fano- Al- challenging task. gebraic Diagrammatic Construction (Fano-ADC) method30–32 Among the resonance phenomena, the Interatomic provides reliable partial ICD widths which are necessary Coulombic Decay (ICD) effect has attracted considerable for full theoretical description of the process. This ab attention in the last decade. ICD is an ultrafast non-radiative initio method is, however, computationally expensive and electronic decay process for excited atoms or molecules not applicable to really large systems. embedded in a chemical environment, like in a cluster. Via We report here on a simpler approach based on this process, the excited species can get rid of the excess the diatomics-in-molecules (DIM) technique33 for the energy, which is transferred to one of the neighbors, and computations of the ICD widths in rare-gas clusters. In the ionizes it. The ICD process was predicted in the late 1990s by standard DIM approach, the Hamiltonian matrix elements of Cederbaum et al.3 It was experimentally demonstrated about a system are evaluated using the energies of the atoms and 10 years ago on the example of neon clusters by Marburger all pairs of atoms forming the system. The DIM Hamiltonian et al.4 and Jahnke et al.5 Since then, it was shown that ICD is matrix is generally small and scales linearly with the number a general process, taking place in a large variety of systems, of atoms within the system. Furthermore, it requires only like hydrogen bonded6,7 or van der Waals clusters.8–11 It has accurate energies of atoms and diatomic molecules which can been observed mainly after photoionization or photoexcitation nowadays be obtained with advanced and reliable quantum but has also been experimentally demonstrated after an chemistry methods. DIM methods have been successfully electron12 and ion impact.13–15 ICD was first predicted and employed for describing small molecules34,35 as well as pure observed after ionization in the inner-valence shell, but it and doped rare-gas clusters.36–39 Here, in the spirit of the was also demonstrated (i) after two-electron processes like DIM, we propose to compute the ICD widths of rare-gas simultaneous ionization-excitation16,17 and double ionization clusters from the widths of each pair of atoms forming the or excitation18,19 and (ii) in cascades after Auger20–23 and cluster. resonant Auger.24,25 In SectionII, we give the general formulas for the The energy width of a resonant state is an important computation of resonance widths from the projection-operator property to characterize the resonance. General quantum formalism. The standard DIM approach for the calculation of mechanical equations for computing the decay widths are potential energy surfaces is then presented. The DIM-based known but are only applicable to small systems. The main method proposed here for the computation of the ICD widths difficulty stems from the fact that one has to account both is finally reported. In Section III, we apply this method to for the many-body and for the scattering aspects of the decay the helium trimer and derive explicit DIM formulas for this system. ICD widths from the DIM approach are compared to a)[email protected] fully ab initio (Fano-ADC) results. 0021-9606/2016/144(8)/084111/7/$30.00 144, 084111-1 © 2016 AIP Publishing LLC Reuse of AIP Publishing content is subject to the terms: https://publishing.aip.org/authors/rights-and-permissions. Downloaded to IP: 195.113.23.13 On: Thu, 25 Feb 2016 16:39:07 084111-2 Sisourat et al. J. Chem. Phys. 144, 084111 (2016) II. ICD WIDTHS IN THE PROJECTION-OPERATOR-DIM Each fragment part Hˆ α contains all kinetic energy operators FRAMEWORK and intra-atomic potential energy terms in Hˆ e which depend only on the coordinates of the electrons assigned originally to The general formulas for the calculation of the ICD α Hˆ αβ widths are derived from the theory of resonances of Fano atom . The operator contains all kinetic energy terms as and Feshbach.40,41 The wavefunctions and coupling matrix well as intra-atomic and inter-atomic potential energy terms α β elements involved in such calculations are then computed associated to the diatomic molecule . within the DIM approach. The main advantage of the latter The DIM basis set is then constructed as antisymmetrized and spin-adapted products of the atomic eigenfunctions χα is that only energies and coupling matrix elements of the m with energy eα of Hˆ α, | ⟩ atomic and diatomic fragments are needed. Computations of m resonance widths in the case of electron-molecule collisions YN anti ˆ ˆ n have been performed with such combined approach (so- Ψm = A Ψm = Asˆ χm : (6) called generalized diatomic-in-molecules approach in this | ⟩ | ⟩ n=1 | ⟩ context).43,44 In the present study, we report a similar approach The operator Aˆ ensures the antisymmetrization ands ˆ creates applied to the calculations of ICD widths. spin-adapted linear combination of the atomic eigenfunction products. In order to evaluate the Hamiltonian matrix elements in the DIM basis set, the diatomic Hamiltonians Hˆ αβ are A. The projection-operator approach αβ expressed via their eigenvalues ϵ i and eigenfunctions A general description of the Fano-Feshbach theory of αβ { } i , resonances within the projection-operator approach is given in {| ⟩} X ˆ αβ αβ αβ αβ Ref. 42. We summarize here the results for a single resonance H = i ϵ i i : (7) state. The generalization for a non-interacting manifold of i | ⟩ ⟨ | resonance states is straightforward. The resonance is described Noting that Aˆ and Hˆ αβ commute, the operation of Hˆ αβ on by a discrete state Ψd , represented by a square integrable Ψanti is simply | ⟩ m wave function, with the energy expectation value given | ⟩ X ˆ αβ anti ˆ ˆ αβ ˆ αβ αβ αβ by H Ψm = AH Ψm = A Ψi ϵ i Ψi Ψm ; (8) | ⟩ | ⟩ i | ⟩ ⟨ | ⟩ ˆ Ed = Ψd He Ψd ; (1) Ψαβ ⟨ | | ⟩ where i is the spin-adapted product of the diatomic | ⟩ αβ where Hˆ e is the full electronic Hamiltonian. This discrete state wavefunction i and the wavefunctions of electrons that | ⟩ αβ is embedded into and coupled to a continuum of final states, are not included in Hˆ , Φ Ψ Y which can be written as f k = f k . Here, the decay Ψαβ αβ n | ⟩ | ⟩| ⟩ i = sˆ i χi : (9) channel Ψf is the eigenstate of the system after the decay | ⟩ | ⟩ | ⟩ | ⟩ n,α; β with energy Ef and k is a single-electron continuum state. | ⟩ αβ αβ Neglecting the inter-channel coupling, the coupling between The overlap element Ψi Ψm = Bim can generally be ⟨ | ⟩ 34,36 the decaying state Ψd and the final state Φf k is determined by symmetry consideration.
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